Asymptotics for Small Nonlinear Price Impact: A PDE Homogenization Approach to the Multidimensional Case

Bayraktar, Erhan; Cayé, Thomas; Ekren, Ibrahim

doi:10.2139/ssrn.3287099

Cited by 6 publications

(7 citation statements)

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“…The expansion provided in this paper allows one to compute the price impact coefficient in the option's market (which is hard to measure directly, due to the lack of liquidity and/or transparency) in terms of the price impact coefficient 𝜂 in the underlying market (which is easier to measure directly), assuming the latter is small-this connection is given explicitly by equation (71). Unlike the existing literature (Bayraktar et al, 2018;Bichuch & Shreve, 2013;Caye et al, 2018;Ekren & Muhle-Karbe, 2019;Guasoni & Weber, 2015;Kallsen & Muhle-Karbe, 2015;Moreau et al, 2017;Possamai et al, 2015), where the authors obtain expansions for the value function of the optimal hedging problem, to obtain the small impact expansion of the indifference price we need to expand a partial derivative of the value function. As the existing methods are not sufficient to obtain such an expansion, we employ a more direct approach that relies on the properties of the optimal control and on the stochastic representations of the derivatives of the value function, established in the preceding part of the paper.…”

Section: Introductionmentioning

confidence: 99%

“…Note that, in this regime, the function 𝑚 is large and, thanks to (B.2), the process (𝜋 * + 𝑄𝜕 𝑠 𝑃), which is the optimally controlled deviation from the frictionless hedge 𝑄𝜕 𝑠 𝑃 𝑡 , is strongly mean reverting around zero. The process (𝜋 * 𝑡 + 𝑄𝜕 𝑠 𝑃 𝑡 )∕𝜂 1∕4 is, in fact, the so called fast variable mentioned in (Bayraktar et al, 2018;Moreau et al, 2017;Soner & Touzi, 2013). However, unlike the latter papers, herein we do not use the viscosity solution methods to characterize the limiting behavior of (𝜋 * 𝑡 + 𝑄𝜕 𝑠 𝑃 𝑡 ).…”

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confidence: 99%

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Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact

Ekren

Nadtochiy

2021

Mathematical Finance

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In this paper, we construct the utility-based optimal hedging strategy for a European-type option in the Almgren-Chriss model with temporary price impact. The main mathematical challenge of this work stems from the degeneracy of the second order terms and the quadratic growth of the first-order terms in the associated Hamilton-Jacobi-Bellman equation, which makes it difficult to establish sufficient regularity of the value function needed to construct the optimal strategy in a feedback form. By combining the analytic and probabilistic tools for describing the value function and the optimal strategy, we establish the feedback representation of the latter. We use this representation to derive an explicit asymptotic expansion of the utility indifference price of the option, which allows us to quantify the price impact in options' market via the price impact coefficient in the underlying market.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact

Ekren

Nadtochiy

2021

Mathematical Finance

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“…Now we would like to show the boundedness of g , which follows the same idea as [8]. Since g is odd, we only need to show that for x > 0, g is bounded from below.…”

Section: A Proof Of Lemma 33mentioning

confidence: 99%

“…Portfolio choice problems for the most tractable specification G(x) = λx 2 /2, λ > 0 are analyzed in singleagent models by [22,3,42,26]; equilibrium returns are determined in [22,43,9]. In [27,14,8], single-agent models are solved for the more general power costs G(x) = λ|x| q /q, q ∈ (1, 2] proposed by [1]. Below, we will determine equilibrium returns for general smooth convex cost functions G as studied in the duality theory of [25]: (ii) The derivative G is also strictly increasing and differentiable on (0, ∞) with G (0) = 0;…”

Section: Costs and Strategiesmentioning

confidence: 99%

Asset Pricing with General Transaction Costs: Theory and Numerics

2019

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We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, the equilibrium returns mean-revert around their frictionless counterparts -the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known wellposedness results, but can be solved numerically using the simulation-based deep-learning approach of [28]. In a calibration to time series of returns, bidask spreads, and trading volume, transaction costs substantially affect equilibrium asset prices. In contrast, the effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.Mathematics Subject Classification: (2010) 91G10, 91G80, 60H10.JEL Classification: C68, D52, G11, G12.

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“…(2018). In Guasoni and Weber (2020); Cayé, Herdegen, and Muhle‐Karbe (2020); Bayraktar, Cayé, and Ekren (2020), single‐agent models are solved for the more general power costs

G false(x false) = {λ | x |}^{q} / q

q \in (1, 2]

proposed by Almgren (2003).…”

Section: Equilibrium With Costs On the Trading Ratementioning

confidence: 99%

Asset pricing with general transaction costs: Theory and numerics

Gonon¹,

Muhle‐Karbe

Shi

2021

Mathematical Finance

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We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, we prove that the equilibrium returns mean-revert around their frictionless counterparts-the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known well-posedness results, but can be solved numerically using the simulation-based deep-learning approach of Han, Jentzen, and E (2018). In a calibration to time series of prices and trading volume, realistic liquidity premia are accompanied by a moderate increase in volatility. The effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.

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Asymptotics for Small Nonlinear Price Impact: A PDE Homogenization Approach to the Multidimensional Case

Cited by 6 publications

References 50 publications

Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact

Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact

Asset Pricing with General Transaction Costs: Theory and Numerics

Asset pricing with general transaction costs: Theory and numerics

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